voting and elections - creicrei.cat/wp-content/uploads/2016/07/lectures-08_01_2010.pdf ·...

Voting and ElectionsPolitical Economics: Week 1

Giacomo A. M. Ponzetto

CREI � UPF � Barcelona GSE

8th and 11th January 2010

Giacomo Ponzetto (CREI) Political Economics 8 - 11 January 2010 1 / 43

Voting and Elections Social Choice

A Policy Problem

A set V of individuals i = 1, ..., I .Individual i has utility function

U�c i , q, p; αi

�αi denotes his exogenous idiosyncratic characteristics;

c i is his private choice subject to constraints H�c i , q, p; αi

�� 0;

q is the policy choice from a feasible set Q;

p = P (c , q) describes the general equilibrium that results from thepolicy choice q and the individual choices c of all agents.



Individual Policy PreferencesIn a well-behaved model, for each policy choice q 2 Q there is aunique equilibrium with actions c (q) and market outcomesp (q) = P (c (q) , q) such that for each agent i 2 V

c i (qjα) = argmaxγU�γ, q, p (q) ; αi

�s.t. H

�γ, q, p (q) ; αi

�� 0.

With price-taking atomistic agents, the individual choice c i does nota¤ect p. Having strategic market interactions makes no di¤erence aslong as there is a unique equilibrium.

The equilibrium yields each agent�s indirect utility function

W�q; αi

�� U

�c i (q) , q, p (q) ; αi

�.

The preferred policy or bliss point of each agent is

q�αi�= argmax

qW�q; αi

�.



Preference Aggregation in GeneralLet Q be a set with at least three alternatives and R the set of allcomplete and transitive (i.e., �rational�) preference relations on Q.Each agent i has a rational preference relation %i2 R.A social welfare functional is a function F : RI ! R that assigns arational social preference relation %= F (%1, ...,%I ) 2 R to anypro�le of rational individual preference relations (%1, ...,%I ) 2 RI .

A social welfare functional F is Paretian if for any pair of alternativesfq, q0g � Q and any preference pro�le (%1, ...,%I ) 2 RI

if q %i q0 for all i , then q % q0.

A social welfare functional F satis�es independence of irrelevantalternatives if for any pair of alternatives fq, q0g � Q and anypreference pro�les (%1, ...,%I ) 2 RI and (%01, ...,%0I ) 2 RI

if %i jfq,q 0g =%0i jfq,q 0g for all i , then % jfq,q 0g =%0 jfq,q 0g.



Arrow�s Impossibility Theorem

Theorem

Every social welfare functional F : RI ! R that is Paretian and satis�esindependence of irrelevant alternatives is dictatorial: there is an agent hsuch that, for any pair of alternatives fq, q0g � Q and any preferencepro�le (%1, ...,%I ) 2 RI , if q �h q0 then q � q0.

One of the conditions of the theorem is F : RI ! R. Hence:I unrestricted domain: de�ned for all preference pro�les;I transitivity: the social preference relation is itself rational.

What can we do? Restrict preferences or specify institutions.

Preference relations are ordinal. There is much more scope foraggregation of individual preferences into a social welfare function ifwe use cardinal utilities and admit their interpersonal comparison.



Majority Rule and the Condorcet Criterion

1 Direct democracy. The citizens themselves make the policy choices.2 Open agenda. Citizens vote over pairs of policies, such that thewinning policy in one round is posed against a new alternative in thenext round and the set of alternatives is the entire feasible set Q.

3 Sincere voting. Citizen i votes for the alternative that maximizes hisindirect utility W

�q; αi

�.

De�nitionA Condorcet winner is a policy q� that beats any other feasible policy in apairwise vote.

CorollaryIf a Condorcet winner q� exists, it is the unique equilibrium under majorityrule (i.e., under the three assumptions above).



The Condorcet Paradox

A Condorcet winner need not exist.

Three agents i 2 f1, 2, 3g and three choices fqA, qB , qC g.Preferences

qA �1 qB �1 qC ,qB �2 qC �2 qA,qC �3 qA �3 qB .

) Majority-rule cycle:

qA � qB , qB � qC , qC � qA.

Arrow�s impossibility theorem for transitive social preference relations.



Strategic Voting

With sincere voting agents reveal their preferences. Why should they?

Let Q be a set with at least three alternatives and P the set of allrational preference relations �i on Q having the property that no twoalternatives are indi¤erent.

A social choice rule is a function F : P I ! Q that assigns a policyq� = F (�1, ...,�I ) 2 Q to any pro�le of individual preferencerelations (�1, ...,�I ) 2 P I .A social choice rule is manipulable if there exists a pro�le(�1, ...,�I ) 2 P I and an agent i such that

F��1, ...,�0i , ...,�I

��i F (�1, ...,�i , ...,�I ) .

If a social choice rule is not manipulable, it is strategy-proof.

A social choice rule is onto if for each q 2 Q there is a(�1, ...,�I ) 2 P I such that F (�1, ...,�I ) = q.



The Gibbard�Satterthwaite Theorem

Theorem

Every social choice rule F : P I ! Q that is onto and strategy-proof isdictatorial: there is an agent h such that F (�1, ...,�I ) �h q for anyq 6= F (�1, ...,�I ) and for any preference pro�le (%1, ...,%I ) 2 P I .

The same result as Arrow�s impossibility theorem.

A social choice rule must be onto to be Pareto e¢ cient, i.e., to pick apolicy when all agents unanimously prefer it to all alternatives.

With an unrestricted domain for preferences, sincere voting istypically inconsistent with strategic rationality.


Voting and Elections The Median Voter

Single-Peaked Preferences

A binary relation � on the set Q is a linear order if it is re�exive,transitive, and total.

De�nitionThe rational preference relation %i is single-peaked with respect to thelinear order � on Q if there is a bliss point qi such that

qi � q > q0 ) q �i q0 and q0 > q � qi ) q �i q0

Concretely, consider Q � R with the natural order � and

W�q; αi

�> W

�q0; αi

�if q

�αi�� q > q0 or q0 > q � q

�αi�

The indirect utility function is strictly quasiconcave with a uniquemaximum q

�αi�.



The Median-Voter Theorem

TheoremSuppose that I is odd and that the preferences of every agent aresingle-peaked with respect to the same linear order on Q. Then aCondorcet winner always exists and coincides with the median-ranked blisspoint qm . It is the unique equilibrium under majority rule.

A simple separation argument.

Since I is odd, the median bliss point qm is uniquely de�ned.

In a pairwise comparison of qm and any q0 > qm , all agents i withbliss point qi � qm > q0 prefer qm . By the assumption of sincerevoting, a majority votes for qm .

Identically for any q00 < qm .



Strategy-Proofness

Let � be a linear order on Q, and P� � P the set of all strict rationalpreference relations on Q that are single-peaked with respect to �.

Theorem

If I is odd there exists a social choice rule F : P I� ! Q that is onto andstrategy-proof and always selects the median-ranked bliss point qm .

Each agent reports a bliss point r i and the median rm is selected.

Truthful reporting r i = qi is a weakly dominant strategy.

For any pro�le of reports r�i by all agents other than i :1 if qi is median in

�qi , r�i

�, i reaches his bliss point;

2 if the median in�qi , r�i

�is rm > qi , an untruthful report could make

it increase but not decrease; symmetrically for rm < qi .

) Thus i can never gain by an untruthful report r i 6= qi .



The Single-Crossing Condition

De�nitionA preference pro�le (%1, ...,%I ) satis�es the single-crossing condition withrespect to the linear order � on Q if

for q > q0 and j > i , q %i q0 ) q %j q0.

Single-peakedness is a property of an individual preference relation.Single-crossing is a property of an entire preference pro�le.

A linear order of voters as well as policies is required.

Concretely, consider Q � R and αi 2 R with the natural order �: ifq > q0 and α0i > αi , then

W (q; αi ) � W�q0; αi

�) W

�q; α0i

�� W

�q0; α0i

�.



The Median Voter Theorem (Second Version)

TheoremSuppose that I is odd and that the agents�preference pro�le satis�es thesingle-crossing condition with respect to the linear order � on Q. Then aCondorcet winner always exists and coincides with the bliss point of themedian agent. It is the unique equilibrium under majority rule.

Proof by the same separation argument as before.

The single-crossing property implies an ordering of bliss pointsα0i > αi ) q (α0i ) > q (αi )

Strategy-proof mechanism: each agent reports a type αi and theenacted policy is the bliss point of the median report q (αm).



Intermediate PreferencesDe�nitionAgents in the set V have intermediate preferences on Q if their indirectutility function can be written as

W (q; αi ) = J (q) +K (αi )H (q) ,

where K (αi ) is monotonic and H, J, and K are common to all agents.

TheoremSuppose that I is odd and that agents have intermediate preferences on QThen a Condorcet winner always exists and coincides with the bliss pointof the median agent. It is the unique equilibrium under majority rule.

Strong restriction, but simple and occasionally convenient.Project on the multidimensional policy space Q the natural orderingof the one-dimensional agent type αi .



Downsian Electoral Competition

Timeline:

1 Two candidates A and B simultaneously and non-cooperativelychoose electoral platforms qA and qB .

2 An election is held in which all citizens vote for either candidate.3 The winner implements his electoral platform (binding commitment).

Voters�strategy:

If qA 6= qB , sincere voting is a weakly dominant strategy.If qA = qB , assume that citizens vote randomly for either candidate.



Politicians�Objectives

O¢ ce-seeking politicians: exogenous �ego rent� from winning.

Maximize the probability of winning pA or pB = 1� pA.Tie breaking assumption qA = qB ) pA = pB = 1

2 .

The result is robust to the assumption that politicians have goalsbeyond winning, so long as they are conditional on winning: e.g.,implementing a certain policy.



Downsian Convergence

TheoremSuppose that two politicians contest an election by announcing a bindingpolicy proposal, and a set of voters V vote for either party following aweakly dominant strategy, and voting randomly when the two proposalsare identical. Suppose that If the voters�preference pro�le on Q is suchthat a Condorcet winner qm exists, there is a unique equilibrium in whichboth parties propose qm .

The median-voter theorem with two-party competition.

Competition on an ordered line à la Hotelling (1929).

Not robust to an increase in the number of parties.



Redistributive Taxation

Agents have preferences over consumption ci and leisure xi :

U (ci , xi ) = ci + V (xi ) ,

where V (xi ) is a well-behaved increasing and concave function.

Policy instrument: a linear tax τ on earnings that is rebated viauniform lump-sum transfers f .

Budget constraint: ci � (1� τ) li + f .

Time-allocation constraint: xi + li � 1+ αi .

Individual productivity αi and labour supply

l (τ; αi ) = argmaxlif(1� τ) li + f + V (1+ αi � li )g

= 1+ αi � V 0�1 (1� τ) .



General Equilibrium

Unit measure of individuals.

αi has a known distribution with mean α and median αm .

Aggregate labour supply

L (τ) = 1+ α� V 0�1 (1� τ)

such thatl (τ; αi ) = L (τ) + αi � α.

Government budget constraint: f � τL (τ).



Policy Preferences

Indirect utility

W (τ; αi ) = (1� τ) l (τ; αi ) + f + V (1+ αi � l (τ; αi ))= (1� τ) (αi � α) + L (τ) + V (1+ α� L (τ)) .

The single-crossing condition is satis�ed.

Individual preferences

∂

∂τW (τ; αi ) = �l (τ; αi ) +

∂f∂τ= α� αi| {z }

redistribution

+ τL0 (τ)| {z }ine¢ ciency

.

Ideal policyτ (αi ) such that τi

��L0 (τi )�� = α� αi



Income Redistribution by the Median Voter

A voter with average productivity (αi = α) wants no redistribution.

) By de�nition, this is the utilitarian welfare optimum.

) With quasilinearity, τ = 0 is also clearly e¢ ciency-maximizing.

Voters with less than average productivity (income) desire progressiveredistribution.

I Voters with more than average income desire regressive redistribution:a production subsidy τ < 0 �nanced by a poll tax.

The median voter prefers τm such that τm jL0 (τm)j = α� αm .

Redistribution increases as the gap between average and medianincome increases. Not any measure of income inequality.

Empirical support for this simple model is not very strong.



Empirical Evidence on the Downsian Model

Gerber and Lewis (2004) focus on elections in Los Angeles County,which contains 55 (state and federal) majoritarian electoral districts.

Voter preferences are measured from a database of 2.8m individualballots for the 1992 election. These votes include 13 statewide ballotpropositions (direct democracy) as well as candidate races.

Ideology of the elected representative is measured from legislativevoting records.

Representatives�ideology is correlated with the median constituent�s,but ...

�Legislators from heterogeneous districts often take policy positionsthat diverge substantially from the median voter in their district.�


Voting and Elections Probabilistic Voting

Beyond Condorcet Winners

The Downsian model fails without a Condorcet winner, and istypically not applicable to multidimensional policy choices.

The platform-choice game lacks a pure-strategy equilibrium.

Discontinuous payo¤ functions and best responses are the problem.

Introduce smoothness by making candidates uncertain about votersupport.

An intensive margin: a voter is the more likely to support a candidate(instead of the competitor, or instead of abstaining), the more he likeshis platform.

Exploit the cardinal dimension of preferences.



Political Preferences

Each candidate P 2 fA,Bg is characterized by the binding platformsqP but also by exogenous non-policy characteristics.

Voters i�s utility if candidate P wins the election is

W�qP ,P; αi

�= W

�qP ; αi

�+ ξPi ,

where ξPi is a stochastic ideological bias.

Let ξBi � ξAi = σi + δ.

σi is an idiosyncratic shock that makes i�s vote imperfectly predictable

δ is a common shock that makes the election imperfectly predictableeven with a continuum of voters.



Probabilistic Voting

Unit measure of voters.

A fraction λj belongs to group j with:1 homogeneous policy preferences αj ;2 i.i.d. ideology σi with distribution Φj (σi ).

Given δ, the fraction of group j that votes for A is

Φj

�W�qA; αj

��W

�qB ; αj

�� δ�.

Aggregate popularity δ is independently drawn from an absolutelycontinuous distribution F (δ).



The Tractable Speci�cation

AssumptionThe idiosyncratic ideology parameters are uniformly distributed:

σi s U"� 12φj

,12φj

#for all voters i in group j .

Given δ, candidate A�share of the vote is

πA (δ) =12+

J

∑j=1

λjφj

hW�qA; αj

��W

�qB ; αj

�i�

J

∑j=1

λjφjδ.

Assume φj and F (δ) such that in each group there are voterssupporting both candidates.

Within these bounds, a group-speci�c mean bias is irrelevant.



Weighted Utilitarian Welfare Function

The probability that candidate A wins the election is

Pr�

πA (δ) >12

�= F

∑Jj=1 λjφj

�W�qA; αj

��W

�qB ; αj

��∑Jj=1 λjφj

!.

O¢ ce-seeking candidates choose

q� = argmaxq

J

∑j=1

λjφjW (q; αj ) .

Policy proposals cater to swing voters: higher φj means that moregroup members are swayed by a policy change.



Another Speci�cationAssumption: Candidates maximize their expected vote share.Candidate A�s expected vote share is

E [πA (δ)] =J

∑j=1

λjEhΦj

�W�qA; αj

��W

�qB ; αj

�� δ�i.

If a symmetric pure-strategy Nash equilibrium of the policy-proposalgame exists, it satis�es

J

∑j=1

λjEhφj (�δ)

irW (q�; αj ) = 0.

In particular if voters�preferences are uncorrelated (δ � 0):

q� = argmaxq

J

∑j=1

λjφj (0)W (q; αj ) .

But a symmetric pure-strategy Nash equilibrium does not typicallyexist with distributions other than the uniform.



Local Public Goods

Individuals in group j have unit income and preferences overconsumption cj and a group-speci�c public good gj :

U (cj , gj ) = ci +H (gj ) ,

where H (gj ) is a well-behaved increasing and concave function.

Policy instrument: provision of public goods g �nanced by a uniformtax τ.

Government budget constraint: ∑Jj=1 λjgj � τ.

Indirect utility

Wj (g) = 1�J

∑i=1

λigi +H (gj )



Preference Aggregation

Utilitarian welfare maximization:

g� = argmaxg

J

∑j=1

λjWj (g)) H 0�g �j�= 1 for all j .

Individual preferences

∂Wj

∂gj= H 0 (gj )� 1| {z }

e¢ ciency

+ 1� λj| {z }redistribution

,

and∂Wj

∂gi= �λj|{z}

redistribution

for all i 6= j .



Public-Goods Provision with Probabilistic Voting

Political support:

g = maxg

J

∑j=1

λjφjWj (g)) H 0 (gj ) =φ

φjfor all j ,

where φ = ∑Jj=1 λjφj is the average density across groups.

If all groups are identically motivated by ideology (φj = φ for all j)electoral competition with probabilistic voting implements theutilitarian social optimum.

There is no clear bias to the size of government. Some groups getmore, some less than g �j .

�Director�s Law� if the middle class is the less ideological group.



Group Size and Political In�uence

Persson and Tabellini (2000) highlight that λj has no direct e¤ect.

However, λj has an indirect e¤ect through its impact on the mean φ.

Consider a change in λj that does not a¤ect the relative sizes of theother groups.

Then the average density across groups i 6= j is a constant φ�j .

In equilibrium, gj depends on λj , φj and φ�j according to

H 0 (gj )� 1 = (1� λj )

φ�jφj� 1!.

Smaller groups have larger policy distortions:I a favoured group bene�ts from being smaller;I a neglected group bene�ts from being larger.



Empirical Evidence on the Power of Swing Voters

Strömberg (2008) studies the allocation of campaign visits acrossstates in a U.S. presidential campaign.

A sophisticated model of the Electoral College system usingprobabilistic voting.

Structural estimation of the underlying parameters based on past voteshares of the parties in each state.

Actual campaign visits line up quite well with theoretical predictions.I This seems to be true for the 2008 election as well.

However, Strömberg (2008) does not consider policy outcomes.

Larcinese, Snyder and Testa (2008, wp) do, and �nd no evidence thatfederal funds are disproportionately allocated to states with moreswing voters or more evenly matched partisan groups.


http://people.su.se/~dstro/WH2008.html

Voting and Elections Voter Turnout

Why Do People Vote?

We routinely assume that all citizens vote. The assumption isstrikingly counterfactual, particularly in the U.S.

Turnout was 62.7% in the 2008 federal election, and only 39.8% forthe �o¤-year�2006 congressional election, which did not feature apresidential race.

In Spain, turnout was 73.85% in the 2008 national election, and only44.9% in the 2009 election for the European Parliament.

Turnout also varies widely across demographic and socioeconomicgroups within a country. It displays huge diversity across countriesand some variation over time, notably with a downward trend inWestern democracies over the past sixty years.

Why do so many people not exercise their right to vote?

Why do so many people bother to vote?



The Paradox of the Rational Voter

The fundamental rational-choice explanation is that people vote toin�uence the outcome of the election.

Going to the polls require time and a modicum of e¤ort, so it has acost κi .

A �rational voter� should vote if and only if his participation issu¢ ciently likely to a¤ect the outcome:

E [Wi (q) ji ]�E [Wi (q) j:i ] � κi .

The left-hand side is positive if the parties would implement di¤erentpolicies.

The magnitude of the left-hand side equals the probability that i�ssingle vote decides the election.

The probability is vanishingly small with millions of ballots cast.



Pivotal-Voter Models

Consider an election with two alternatives qA and qB .

nA voters have welfare WA (q) such that WA (qA)�WA (qB ) = 1.

nB have welfare WB (q) such that WB (qB )�WB (qA) = 1.

The policy is determined by the toss of a fair coin if the election endsin a tie.

Voter i is pivotal and gains 12 from voting in two cases:

1 When the number of ballots cast by other voters for either party isidentical.

2 When the number of ballots cast for the opposing party is exactly onemore than that of ballots cast by other voters for i�s own party.



Mixed-Strategy Equilibria

Suppose that all voters have an identical cost κ > 0 of casting aballot.

There cannot be an equilibrium in which all voters abstain so long asκ < 1

2 .

Equilibria in pure strategies typically fail to exist.

Mixed-strategy equilibria always exist, and are not unique.

1 Symmetric equilibria in which all voters in the same group vote withequal probability.

2 Asymmetric equilibria in which some voters randomize and other playeither pure strategy.



Symmetric EquilibriumEvery A supporter votes with probability pA and every B supportervotes with probability pB .In equilibrium pA and pB are such that all citizens are indi¤erent:

Pr (B (nB , pB )� B (nA � 1, pA) = 0)+ Pr (B (nB , pB )� B (nA � 1, pA) = 1) = 2κ

Pr (B (nA, pA)� B (nB � 1, pB ) = 0)+ Pr (B (nA, pA)� B (nB � 1, pB ) = 1) = 2κ,

where B (n, p) denotes a binomial distribution, and independence ofthe two binomials is implicit.Palfrey and Rosenthal (1983) show that as n = nA + nB grows large,pA and pB tend to zero, and so does expected turnout.If nA = nB there is also an exceptional equilibrium with pA = pB � 1,but the standard equilibrium with pA = pB � 0 continues to exist.



An Asymmetric Equilibrium

Let nA > nB .

All members of the majority play a pure strategy, and exactly nB goto the polls.

Each member of the minority votes with independent probability p

Minority members are indi¤erent if pnB�1 = 2κ.

Majority voters do not wish to abstain ifpnB + nB (1� p) pnB�1 � 2κ.

Majority abstainers do not wish to vote if pnB � 2κ.

) The equilibrium exists for p = (2κ)1

nB�1 .

Expected turnout can be arbitrarily close to one, for nA � nB :

nBnA + nB

h1+ (2κ)

1nB�1

i! 2nBnA + nB

as nB ! ∞.



Bayesian Nash EquilibriumLet the cost of voting κi be i.i.d. with an absolutely continuousdistribution F (κi ) on [κ, κ].There is a symmetric equilibrium in which each A supporter votes ifand only if κi < κ�A and each B supporter if and only κi � κ�B .In equilibrium κ�A and κ�B are such that all citizens are indi¤erent:

Pr (B (nB ,F (κ�B ))� B (nA � 1,F (κ�A)) = 0)

+ Pr (B (nB ,F (κ�B ))� B (nA � 1,F (κ�A)) = 1) = 2κ�A

Pr (B (nA,F (κ�A))� B (nB � 1,F (κ�B )) = 0)

+ Pr (B (nA,F (κ�A))� B (nB � 1,F (κ�B )) = 1) = 2κ�B .

Palfrey and Rosenthal (1985) show that if [0, 1] � [κ, κ] then κ�A andκ�B tend to zero as the electorate grows large.

) If someone certainly votes and someone certainly abstains, peopleonly vote for non-strategic reasons, i.e., when κi < 0.



Bene�ts of VotingThe standard calculus of voting allows not only for a cost of going tothe polls but also from a bene�t of voting.

The simplest assumption is that people engage in expressive votingand derive a psychological bene�t ψ (Wi (qA)�Wi (qB )) fromsupporting A against B.

I Turnout rises with the stakes of the election.I Candidates�incentives to chose platforms are analogous to those of theprobabilistic-voting model, and as tractable if κi s U [κ, κ].

Group-level theories of turnout do not su¤er from the rational-voterparadox:

I Each voter�s turnout generates positive externalities for like-mindedindividuals.

I If externalities within the group are internalized, members follow hejointly optimal rule, which involves much higher turnout than theindividually optimal behaviour.

I Turnout rises in both the stakes and the closeness of the race.



Microfoundations of Group Participation1 Shachar and Nalebu¤ (1999) suggest that politicians and otheropinion leaders invest resources in increasing their followers�bene�t ofvoting (or reducing the cost).

2 Feddersen and Sandroni (2006) appeal to the ethics of ruleutilitarianism.

I Ethical agents follow the voting rule that would maximize social welfareif it were followed by all agents, while non-ethical agents abstain.

I There are two kinds of ethical agents, who disagree over theassessment of the social-welfare consequences of the two policies.

3 Coate and Conlin (2004) give the argument a twist with group ruleutilitarianism: each agent only cares about the welfare of members ofhis own group.

I They �nd this model outperforms simple expressive voting in explainingparticipation in Texas liquor referenda.

I Coate, Conlin and Moro (2008) in turn �nd that the expressive votingmodel outperforms the pivotal-voter model.


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